Physics307L:Schedule/Week 9 agenda/Weighted

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The problem: You have two separate measurements with different uncertainties. How do you combine them?

Example: Speed of Light

Measurement #1 (25 +/- 3) cm / ns

Measurement #2 29(2) cm / ns

What is the best way to combine these measurements?

What do you expect the new means, uncertainties to be?

Derivation

The formula

Wikipedia article

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \bar{x} = \frac{ \sum_{i=1}^n w_i x_i}{\sum_{i=1}^n w_i}, }

which means:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \bar{x} = \frac{w_1 x_1 + w_2 x_2 + \cdots + w_n x_n}{w_1 + w_2 + \cdots + w_n}. }
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle w_i = \frac{1}{\sigma_i^2}. }

Excel examples